Question : If $x+\frac{1}{x}=1$, then the value of $\frac{x^2+7x+1}{x^2+11x+1}$:
Option 1: $\frac{3}{4}$
Option 2: $\frac{2}{3}$
Option 3: $\frac{1}{3}$
Option 4: $\frac{1}{4}$
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Correct Answer: $\frac{2}{3}$
Solution :
Given: $x+\frac{1}{x}=1$
Dividing numerator and denominator of $\frac{x^2+7x+1}{x^2+11x+1}$ by $x$.
So, $\frac{x^2+7x+1}{x^2+11x+1}$ = $\frac{x+7+\frac{1}{x}}{x+11+\frac{1}{x}}$
= $\frac{1+7}{1+11}$
= $\frac{8}{12}$
= $\frac{2}{3}$
Hence, the correct answer is $\frac{2}{3}$.
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